2.4 CAPM and APT

In [3]:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.pylab as pylab
import datetime as dt
from sklearn.linear_model import LinearRegression
from sklearn.decomposition import PCA
from scipy.stats import pearsonr
import warnings
warnings.filterwarnings('ignore')
In [4]:
params = {'figure.figsize': (15, 5),
         'axes.labelsize': 14,
         'axes.titlesize': 14,
          'axes.titlesize': 18}
pylab.rcParams.update(params)

2.4.1

In [5]:
df = pd.read_csv(r'/Users/desmond/Desktop/FSP_CW/all/Section 2/data/fsp_case_31_BSD.csv',index_col=0,header=[0,1])
df.fillna(value=0,inplace=True)
#returns dataframe
df_returns = df['ret']
date_axis = [dt.datetime.strptime(d,'%Y-%m-%d').date() for d in df_returns.index]
df_returns.shape
Out[5]:
(521, 157)
In [6]:
a = df_returns.rolling(22,min_periods=22).mean().dropna()
In [7]:
rmt = np.mean(df_returns,1)
plt.figure()
plt.plot(date_axis,rmt)
plt.grid('major',axis='both')
plt.xlabel('Date')
plt.ylabel('Simple returns')
plt.title('Daily market returns averaged across 157 European companies')
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/241.eps')

2.4.2

In [8]:
#Beta = Cov(Ri,Rm) / Var(Rm)
win = 22
betas = np.zeros((len(df)-win+1,df['ret'].shape[1]))
for i in range(df['ret'].shape[1]):
    for j in range(len(df)-win+1):
        betas[j,i] = df['ret'].iloc[j:win+j,i].cov(rmt.iloc[j:win+j])/rmt.iloc[j:win+j].var()
In [9]:
df['ret']['G_AALLN'].rolling(22).cov(rmt).dropna()
Out[9]:
2017-01-31    0.000027
2017-02-01    0.000030
2017-02-02    0.000030
2017-02-03    0.000023
2017-02-06    0.000027
                ...   
2018-12-25    0.000196
2018-12-26    0.000202
2018-12-27    0.000213
2018-12-28    0.000228
2018-12-31    0.000218
Length: 500, dtype: float64
In [10]:
plt.figure()
plt.plot(df.iloc[21:,:].index,betas)
plt.xticks([df.iloc[21:,:].index[i] for i in np.linspace(0,len(df.iloc[21:,:])-1,10).astype(int)])
plt.xlabel('Date')
plt.ylabel('Beta')
plt.title('Rolling betas for all stocks')
plt.grid('major',axis='both')
plt.tight_layout
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/242.eps')

2.4.3

In [11]:
daily_total_mcap = df['mcap'].sum(axis=1)
mcap_times_return = np.multiply(df['mcap'],df['ret']).sum(axis=1)
daily_weighted_return = mcap_times_return / daily_total_mcap
percentage_return = 100 * daily_weighted_return
plt.figure()
plt.plot(percentage_return)
plt.xticks([percentage_return.index[i] for i in np.linspace(0,len(percentage_return.index)-1,10).astype(int)])
plt.grid('major',axis='both')
plt.xlabel('Date')
plt.ylabel('Return (%)')
plt.title('Daily cap-weighted market returns')
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/243.eps')

2.4.4

In [12]:
#Beta = Cov(Ri,Rm) / Var(Rm)
win = 22
cap_weighted_betas = np.zeros((len(df)-win+1,df['ret'].shape[1]))
for i in range(df['ret'].shape[1]):
    for j in range(len(df)-win+1):
        cap_weighted_betas[j,i] = np.cov(df['ret'].iloc[j:win+j,i],(daily_weighted_return.iloc[j:win+j]))[1,0]/daily_weighted_return.iloc[j:win+j].var()
In [13]:
plt.figure()
plt.plot(df.iloc[21:,:].index,cap_weighted_betas)
plt.xticks([df.iloc[21:,:].index[i] for i in np.linspace(0,len(df.iloc[21:,:])-1,10).astype(int)])
plt.xlabel('Date')
plt.ylabel('Beta')
plt.title('Cap-weighted rolling beta for all stocks')
plt.grid('major',axis='both')
plt.tight_layout
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/244.eps')
In [14]:
betas_pd = pd.DataFrame(betas.T,index=df['mcap'].T.index).T
cap_weighted_betas_pd = pd.DataFrame(cap_weighted_betas.T,index=df['mcap'].T.index).T
In [15]:
plt.figure()
plt.plot(df.iloc[21:,:].index,betas_pd.T.loc['G_NESNVX'])
plt.plot(cap_weighted_betas_pd.T.loc['G_NESNVX'])
plt.xticks([df.iloc[21:,:].index[i] for i in np.linspace(0,len(df.iloc[21:,:])-1,10).astype(int)])
plt.title('Beta comparison: largest-cap stock (G_NESNVX)')
plt.xlabel('Date')
plt.ylabel('Beta')
plt.legend(['Rolling beta','Cap-weighted beta'])
plt.grid('major',axis='both')
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/largestcap.eps')

The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.
The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.

2.4.5 a)

In [16]:
bs = df['mcap'].rolling(22).mean().dropna().apply(np.log).replace([np.inf, -np.inf],0).T
bm = cap_weighted_betas.T
ri = df['ret'].iloc[21:,:].T
In [17]:
def apt_regression(factor,ri,bm,bs):
    coeffs = np.zeros((factor,1))
    a = np.zeros((1,1))
    resid = np.zeros((ri.shape[0],1))
    for i in range(ri.shape[1]):
        X = np.concatenate([[bm[:,i],bs.values[:,i]]]).T
        y = ri.values[:,[i]]
        reg = LinearRegression()
        reg.fit(X,y)
        #print(coeffs.shape)
        coeffs = np.append(coeffs,reg.coef_.T,axis=1)
        a = np.append(a,reg.intercept_)
        resid = np.append(resid,reg.predict(X) - ri.values[:,[i]],axis=1)
    
    Rm = coeffs[0,:]
    Rs = coeffs[1,:]
    pred = np.multiply(Rm[1:],bm)+ np.multiply(Rs[1:],bs) + a[1:]
    resid = ri - pred

    return Rm[1:], Rs[1:], a[1:], resid
In [18]:
factor = 2
Rm, Rs, a, resid = apt_regression(factor,ri,bm,bs)

plt.figure()
plt.plot(df.iloc[21:,:].index,a)
plt.plot(df.iloc[21:,:].index,Rm)
plt.plot(df.iloc[21:,:].index,Rs)
plt.xticks([df.iloc[21:,:].index[i] for i in np.linspace(0,len(df.iloc[21:,:].index)-1, 10).astype(int)])
plt.title('Estimation of market regression coefficients per day')
plt.xlabel('Date')
plt.ylabel('Magnitude')
plt.legend(['a','Rm','Rs'])
plt.grid('major',axis='both')
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/245a.eps')

The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.
The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.

2.4.5 b)

In [19]:
mean_a = np.mean(a)
std_a = np.std(a)
mean_Rm = np.mean(Rm)
std_Rm = np.std(Rm)
mean_Rs = np.mean(Rs)
std_Rs = np.std(Rs)
stats = pd.DataFrame(data=[[mean_a, std_a],[mean_Rm,std_Rm],[mean_Rs,std_Rs]],index=['a','Rm','Rs'],columns=['mean','std'])
In [20]:
stats
Out[20]:
mean std
a -0.000068 0.003663
Rm -0.000306 0.007825
Rs 0.000017 0.000234
In [21]:
print(ri.shape,resid.shape,a.shape,Rm.shape,Rs.shape)

(157, 500) (157, 500) (500,) (500,) (500,)
In [22]:
ri.index
Out[22]:
Index(['G_AALLN', 'G_ABIBB', 'G_ACFP', 'G_ADENVX', 'G_ADLERR', 'G_ADNA',
       'G_AFFP', 'G_AIFP', 'G_AIRFP', 'G_AKZANA',
       ...
       'G_TUIGR', 'G_UNANA', 'G_UPMFH', 'G_VIEFP', 'G_VIVFP', 'G_VLVY',
       'G_VOD', 'G_VW', 'G_WKLNA', 'G_WPPLN'],
      dtype='object', name='cmp_id', length=157)

2.4.5 c)

In [23]:
corr_all = np.zeros((ri.shape[0]))
for i in range(ri.shape[0]):
    #corr, _ = pearsonr(resid.T[:,i],ri.values.T[:,i])
    corr = np.corrcoef(resid.T.iloc[:,i],ri.values.T[:,i])[0,1]
    corr_all[i] = corr
print(corr_all.shape)
plt.figure()
plt.plot(ri.index,corr_all)
plt.xticks([ri.index[i] for i in np.linspace(0,len(ri.index)-1,20).astype(int)],rotation=45)
plt.title('Correlation between specific returns and stock returns per company')
plt.xlabel('Stock')
plt.ylabel('Correlation')
plt.grid('major',axis='both')
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/245c.eps')

(157,)

2.4.5 d)

In [24]:
R_ = np.append(Rm.reshape(500,1),Rs.reshape(500,1),axis=1)
eigR_all = np.zeros((2,len(R_)-win+1))
for i in range(len(R_)-win+1):
    covR = np.cov(R_[i:win+i,:].T)
    eigR, _ = np.linalg.eig(covR)
    eigR_all[:,i] = eigR.T
    
plt.figure(figsize=(20,5))
plt.subplot(121)
plt.plot(eigR_all[0,:])
plt.xlabel('Covariance matrix index')
plt.ylabel('Magnitude')
plt.title('First eigenvalue of each covariance matrix')
plt.grid('major',axis='both')
plt.subplot(122)
plt.plot(eigR_all[1,:])
plt.xlabel('Covariance matrix index')
plt.ylabel('Magnitude')
plt.title('Second eigenvalue of each covariance matrix')
plt.grid('major',axis='both')
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/245d.eps')

2.4.5 e)

In [30]:
cov_e = np.cov(resid.T)
pca = PCA(n_components=61)
pca.fit(resid)
print(pca.explained_variance_ratio_[0:7].sum())

plt.figure(figsize=(10,5))
plt.plot(pca.explained_variance_ratio_)
plt.xlabel('Principal component')
plt.ylabel('% of total variance')
plt.title('PCA of specific return matrix: component variance percentage')
plt.xlim([0,60])
plt.grid('major',axis='both')
#plt.savefig('/Users/desmond/Desktop/FSP_CW/all/Section 2/eps_files/245e.eps')

0.2830590752826653
In [26]:
"""covariance matrix generated from original dataset
eigendecomposition/SVD of covariance matrix
sort eigendecomposition matrices by eigenvalue magnitude
The eigenvalue matrix, by mathematical definition, contains the variance components of dataset feature space
the eigenvector matrix is therefore also sorted by the corresponding eigenvalue magnitude
select n principal components - choose n eigenvectors with the highest variance
Project dataset onto principal components to reduce original dataset dimension

Each principal component contains a percentage of total dataset variance
If 5 principal components contains 80% of the total variance

  File "<ipython-input-26-d83f65f22951>", line 10
    If 5 principal components contains 80% of the total variance
                                                                
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In [ ]: